Langmuir Adsorption Isotherm Simulator — Coverage and Equilibrium Constant
Visualize the monolayer adsorption θ = Kp/(1+Kp). Vary pressure, temperature, heat of adsorption and pre-exponential factor to see the transition from the Henry to the saturation regime.
Parameters
Pressure p
kPa
Temperature T
K
Heat of adsorption |ΔH_ads|
kJ/mol
Pre-exponential factor K_0
1/kPa
K(T) = K_0·exp(|ΔH_ads|/(R·T)). Gas constant R = 8.314 J/(mol·K). ΔH_ads is exothermic (negative); enter its magnitude.
Results
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Coverage θ
—
Equilibrium constant K(T)
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Half-saturation pressure p_50 = 1/K
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Adsorption regime
Langmuir Isotherm and Temperature Comparison
Top: θ vs p at current conditions (log axis, red dot = current point, dashed = θ=0.5 and p_50). Bottom: isotherms at T = 250, 298, 400, 600 K.
Theory & Key Formulas
The Langmuir model assumes equivalent surface sites with monolayer coverage. Equating adsorption and desorption rates at equilibrium yields the isotherm:
Coverage (occupied site fraction). K is the equilibrium constant and p the partial pressure:
$$\theta = \frac{K\,p}{1 + K\,p}$$
Temperature dependence of the equilibrium constant (van't Hoff). R is the gas constant; ΔH_ads is the (negative) heat of adsorption:
Half-saturation pressure (θ = 0.5). Larger K → saturation at lower pressure:
$$p_{50} = \frac{1}{K}$$
At low pressure (Kp ≪ 1) θ ≈ Kp (Henry region); at high pressure (Kp ≫ 1) θ → 1 (saturation). The isotherm shape depends only on the product Kp.
What is the Langmuir Adsorption Isotherm Simulator?
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I have never heard of "adsorption isotherms." Is it some equation for gas molecules sticking onto a solid surface?
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Exactly that. Roughly speaking, hold the temperature fixed, change the gas pressure, and ask "what fraction of the surface sites are occupied?" That relation is the isotherm. Langmuir proposed the simplest one in 1918 and it fits in one line: $\theta = Kp/(1+Kp)$. Try moving the pressure slider above and watch the coverage θ rise from 0 toward 1 in an almost S-shaped curve.
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Why is the horizontal axis logarithmic? Wouldn't a linear scale be easier?
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Good catch. Real isotherms have three faces — Henry, transition, and saturation — and they typically span more than 100x in pressure. On a linear axis the low-pressure region collapses to a thin line. On a log axis the slope at low p (which equals K) and the plateau at high p (θ → 1) both fit on one plot. Sweep p from 0.01 to 1000 kPa above and you'll appreciate the log scale.
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When I raise the temperature, the curves in the bottom panel all march to the right. So heat makes adsorption harder?
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Right. Adsorption is usually exothermic, so by Le Chatelier raising T shifts equilibrium toward desorption. Mathematically $K(T) = K_0\exp(|\Delta H|/RT)$: bigger T shrinks the exponent and shrinks K. Smaller K means a larger half-saturation pressure $p_{50}=1/K$, which is what you see as the right-shift. The same physics is why cryogenic adsorption is so much more effective than room-temperature adsorption.
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When I crank up the heat of adsorption, even tiny pressures give near-full coverage. Is this related to temperature too?
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Very much so. |ΔH| measures the bond strength to the surface — 40-400 kJ/mol for chemisorption, 5-40 kJ/mol for physisorption. Push |ΔH| up to about 40 kJ/mol and K explodes, so θ pins near 1 even at 1 kPa. Catalyst design lives in this trade-off: bond strongly enough to react fast, but weakly enough that products desorb. Tuning ΔH is much of what catalyst research actually is.
FAQ
Chemisorption is monolayer and site-specific, so Langmuir fits well. Physisorption is weak and tends to multilayer, so Langmuir works at low pressure but BET is needed at high coverage. Check the linearised plots (1/θ vs 1/p or p/n vs p): a straight line confirms Langmuir behaviour.
In Langmuir-Hinshelwood kinetics the rate is r = k θ_A θ_B, where coverages come from the isotherm. The rate scales linearly with pressure at low coverage and plateaus near saturation, so the optimal operating pressure is intermediate. Measuring CO or H_2 isotherms on a metal surface is the standard way to set the temperature/pressure window.
Semiconductor gas sensors (e.g. SnO2) change conductivity with surface oxygen coverage. At low concentrations the response is linear (θ ≈ Kp), defining the dynamic range, while at high concentrations the response saturates. Designers tune K (via temperature or catalytic additives) so that the slope at low p (sensitivity) and p_50 (range centre) match the target concentration.
BET extends Langmuir to multilayer adsorption, treating second-and-higher layers like liquid-like adsorption — it underpins the standard N2 surface-area measurement (BET method). Freundlich is the empirical $\theta = K p^{1/n}$, useful for heterogeneous surfaces but unable to model saturation. Langmuir is the idealised reference model for monolayer adsorption on uniform sites.
Real-world applications
Heterogeneous catalyst design and operation: Automotive exhaust converters, ammonia synthesis, methanol reforming — rates depend strongly on surface coverages. Langmuir-type coverages are the input to Langmuir-Hinshelwood rate laws used for temperature, pressure and feed-composition optimisation, reactor sizing and catalyst-life prediction.
Separation and purification by porous adsorbents: Activated carbon, zeolites and MOFs separate gas mixtures based on component isotherms. Pressure-swing adsorption (PSA) and temperature-swing adsorption (TSA) exploit the K(T, p) dependence to cycle adsorption and desorption. Air separation, CO2 capture and hydrogen purification are mature applications.
Gas sensors and environmental monitoring: Semiconductor and electrochemical sensors respond to surface adsorption. The Henry-region slope sets sensitivity; p_50 sets full-scale. Low-concentration ambient pollutants (NOx, CO, VOCs) require careful tuning of K(T) via on-chip heaters.
Pharmaceutical and food moisture isotherms: Drug and food stability depends on water activity, set by water-vapour adsorption equilibria. Langmuir describes the low-activity region; BET takes over at high activity. These isotherms drive storage, packaging and shelf-life decisions throughout formulation development.
Common pitfalls
The most common mistake is to assume the Langmuir model fits any adsorption. It rests on four idealisations — equivalent sites, monolayer, no lateral interactions, coverage-independent ΔH. Real surfaces have steps, kinks and defects, and adsorbates interact attractively or repulsively. At high coverage or with multilayer physisorption, Langmuir systematically deviates from data. If a 1/θ vs 1/p plot is not linear, switch to Freundlich, Temkin or BET.
The next pitfall is ignoring the temperature dependence of K and assuming "one isotherm is enough." Because K varies exponentially with 1/T, a 50 K change can move K by a factor of several to several dozen. Look at the four temperatures (250-600 K) in the lower panel: at the same |ΔH| the curves shift roughly three decades in pressure. Always check K at the actual operating temperature; conversely, several isotherms enable a van't Hoff plot (ln K vs 1/T) whose slope gives ΔH experimentally.
Finally, beware of confusing coverage θ with adsorbed amount n. θ is the dimensionless occupied-site fraction (0-1); n is an extensive amount (e.g. mol/g of adsorbent). They relate as n = n_max · θ, where n_max depends on surface area and site density. Fitting data to a Langmuir form requires first converting n to θ by estimating n_max. The reliability of any Langmuir fit is dominated by how well that total-site count is determined.